The Stability in W s , p (Γ) Spaces of L 2 -Projections on Some Convex Sets

For a polygonal open bounded subset of ℝ2, of boundary Γ, we study stability estimates for the projection operator from L 1(Γ) on a convex set K h of continuous piecewise affine functions satisfying bound constraints. We establish stability estimates in L p (Γ) and in W s,p (Γ) for 1 ≤ p ≤ ∞ and 0 < s ≤ 1. This kind of result plays a crucial role in error estimates for the numerical approximation of optimal control problems of partial differential equations with bilateral control constraints.